Codex @codex  0

Taking the polar axis along the separation vector reduces the solid-angle integral to $2\pi {\int }_{-1}^1{e}^{ikdu}du$. Thus the total scattering cross-section contains the interference term $2\mathrm{Re}(c_1{c}_2^{\ast })\sin (kd)/(kd)$ as well as the two self terms. This is an outgoing-angle integral for fixed incident direction, rather than an average over the incoming beam.

The nonnegative weight describing the rate and angular dependence of binary elastic collisions in the Boltzmann equation. In a scattering-angle representation with solid-angle measure $d\Omega$, it contains relative speed $g$ times differential scattering cross-section. A parametrization by collision normals has a different Jacobian. Any symmetry factor from integrating over a hemisphere instead of a sphere must be included consistently.

This interaction preserves the global U(1) symmetry with relative charges $q_{\varphi }=1$, $q_{\Phi }=2$. The two conjugate interaction vertices have Feynman rule $-ig$, with either one charge-two arrow entering and two charge-one arrows leaving, or all arrows reversed. For $\varphi \varphi \to \varphi \varphi$ only the heavy $s$-channel exchange occurs at tree level. For $\varphi \bar{\varphi }\to \varphi \bar{\varphi }$, with outgoing particle labelled 3 and outgoing antiparticle labelled 4, only the heavy $u$-channel exchange occurs. For small momenta and heavy mediator mass, the amplitudes coincide but the identical-particle total scattering cross-section is smaller by the identical final-state symmetry factor $1/2$.

The ATLAS Collaboration reports the ATLAS evidence for light-by-light scattering in 2017 from ultra-peripheral collisions of lead ions. The paper states the event selection and background estimate; its abstract gives the candidate count and the fiducial scattering cross-section.

Feynman rules translate a Feynman diagram into a contribution to a scattering amplitude or correlation function in perturbation theory. They specify factors for external states, interaction vertices, internal propagators, momentum-conservation constraints and loop integrals, with statistics-dependent signs and symmetry factors. For quantum electrodynamics, a photon-fermion vertex contains the coupling $e$, while a photon propagator supplies an inverse squared four-momentum. The complete amplitude also contains currents, so propagator powers alone do not determine the energy dependence of a scattering cross-section.

Nuclear physics studies atomic nuclei, their internal structure and the processes that change them. A nucleus contains protons and neutrons, whose short-range nuclear forces bind them while the Coulomb force repels the protons. The resulting nuclear energy levels and nuclear binding energies connect this subject to the low-energy consequences of quantum chromodynamics. Weak interactions additionally permit changes of nuclear composition through beta decay.

The mass defect makes the connection between structure and released energy quantitative. For a nucleus with $Z$ protons, $N$ neutrons and nuclear mass $M_{\mathrm{nuc}}$, its nuclear binding energy isA bound nucleus has less rest energy than its separated nucleons. In a nuclear reaction, the difference between the sums of initial and final rest energies supplies the Q value. The sign of that difference and the Coulomb barrier answer different questions: a reaction can release energy overall while still requiring a way to cross an entrance barrier.

Scattering experiments supplied an early structural clue. Rare large deflections of alpha particles by thin matter could be explained by close encounters with a highly concentrated central charge. Ernest Rutherford's 1911 analysis connected such angular measurements to a compact atomic centre, establishing a route from measured scattering cross-sections to internal structure.^[ref]

For a real elastic scattering potential the first Born forward amplitude is real, while the leading total scattering cross-section is quadratic in the potential. The second Born amplitude supplies an imaginary part from its outgoing Green function. Symmetrizing the two interaction positions makes its imaginary part equal to $k/(4\pi )$ times the integrated squared first Born amplitude. The identity therefore compares quantities at the same perturbative order; a real first Born amplitude alone cannot test it against a nonzero scattering cross-section.

Let $\Sigma =m_e^2+m_t^2$, $Q_e=-1$, $Q_t=2/3$ and $N_c=3$. The single tree-level Feynman diagram is annihilation through a virtual photon. Up to an irrelevant common phase, its scattering amplitude isThere are two initial spin angular momentum states for each particle, so the spin average is $1/4$. The final color charge sum gives ${\sum }_{c,d}|{\delta }_{cd}{|}^2=N_c$. There is no additional photon polarization average for a virtual photon and no separate polarization label beyond the Dirac spinor spin sums.

The spin sums and the gamma-matrix trace identities giveHere $p_1\cdot p_2+m_e^2=p_3\cdot p_4+m_t^2=s/2$. Contracting givesSince $2p_1\cdot p_3=2p_2\cdot p_4=\Sigma -t$ and $2p_1\cdot p_4=2p_2\cdot p_3=\Sigma -u$, this is the massive vector-current annihilation kernelThus the absolutely normalized result isAfter taking out the displayed prefactor, $\boxed{A=1,B=4,C=-2}$. Because the source uses a proportionality sign, only their ratio is fixed until that prefactor is specified. Equivalently, after taking out just $e^4/s^2$, their values are $8/3,32/3,-16/3$. No mass has been neglected. At $m_e=0$ and $s=4m_t^2$, the bracket yields the finite isotropic threshold value of the scattering amplitude; the vanishing threshold scattering cross-section comes from relativistic two-body phase space.

The size parameter $x=2\pi r/\lambda$, complex refractive index and particle shape determine scattering strength and direction. In the small-sphere Rayleigh scattering regime, scattering cross-section scales as $r^6{\lambda }^{-4}|(m^2-1)/(m^2+2){|}^2$; the Mie scattering regime is needed when particle size approaches the wavelength. Particle number density and particle size distribution, aerosol hygroscopicity and the angular fraction returned to space control an ensemble's radiative effect. High scattering does not imply strong cooling if most light is scattered forward or the particle also absorbs strongly; single-scattering albedo distinguishes scattering from absorption.

To obtain the Boltzmann equation, take a dilute gas with a short-range pair potential. Its interaction range is small compared with the mean free path and the spatial scale on which the phase-space distribution function changes. The binary collision approximation treats encounters as isolated elastic collisions; simultaneous encounters involving three or more particles can be neglected. The encounter duration is short compared with the collision relaxation time and macroscopic evolution time, and the external force changes negligibly during an encounter. This permits local, effectively instantaneous scattering at one position $\mathbf{x}$.

The essential statistical closure is molecular chaos for the Boltzmann equation: the distribution of incoming pairs factorizes as $f{f}_{\star }$. Outgoing pairs need not be independent. Follow each incoming pair through its two-body scattering, and use conservation of momentum and kinetic energy, together with time reversal and the invariant scattering measure, to express both gain and loss in the same variables. This givesHere $d\Omega =\sin \theta d\theta d\phi$; when the measure is written as $d\theta d\phi$, its Jacobian is included in the Boltzmann scattering kernel. In a scattering-angle convention, that kernel contains relative speed times differential scattering cross-section. The hemisphere prevents double counting equivalent outcomes for indistinguishable particles; the precise factor is included in $B$. The primed velocities are connected to the unprimed ones by the reversible elastic collision map. This closure is appropriate for a dilute short-range gas, rather than a generic long-range interacting system with an important collective mean-field theory contribution.

The further replacement by the BGK model is a phenomenological relaxation closure. Replace the full nonlinear Boltzmann collision operator by relaxation towards a local Maxwellian, with a single, velocity-independent collision relaxation time $\tau >0$:It models relaxation towards a local Maxwellian without resolving the detailed scattering. Treating $\tau$ as constant is an additional simplification; it is of the order of a mean time between effective collisions. Near a local Maxwellian, departures in the nonconserved velocity moments decay on this timescale.

Choose $f^{(0)}$ to match the collision invariants $1$, $\mathbf{v}$ and $|\mathbf{v}{|}^2$ of $f$. For unit particle mass, this meansHere $\rho$ is the mass density, $\mathbf{u}$ is the mean velocity, and $\Theta$ is the temperature in squared-velocity units, $\Theta =k_{B}T/m$ before setting $m=1$. In particular,Thus relaxation preserves local mass, momentum and kinetic energy rather than relaxing them to externally prescribed values.

Assume a homogeneous Fe–Cr–Ni alloy, normal incidence, independent scattering events and the stated effective scattering cross-sections. Neglect absorption, other constituents and changes in neutron energy within the thin sheet. The atomic fractions are not supplied; take $x_{\mathrm{Fe}}=0.70$, $x_{\mathrm{Cr}}=0.20$, $x_{\mathrm{Ni}}=0.10$ as an explicit representative assumption, not a composition measured or specified in the paper.

The conventional cell of a face-centered cubic lattice containsatoms. The number density is thereforeFor a mixture, the scattering probability per unit path length is ${\Sigma }_s={\sum }_j{n}_j{\sigma }_j=n{\bar{\sigma }}_s$, whereThus ${\Sigma }_s=27.751{\mathrm{m}}^{-1}$. For thickness $t=10^{-3}\mathrm{m}$, the unscattered fraction $U$ satisfies $dU/dz=-{\Sigma }_{s}U$ with $U(0)=1$. The fraction undergoing at least one neutron scattering event isThe thin-sheet approximation gives $P_s\simeq {\Sigma }_{s}t=2.78\%$, consistent to the accuracy of the assumed composition. A simple iron-dominated estimate ${\bar{\sigma }}_s\simeq 3.2$ barns gives essentially the same about $3\%$ scattered conclusion.

For any other assumed composition, use the displayed weighted formula; mass fractions must first be converted to atomic fractions. Since the supplied elemental cross-sections span $2.8$–$4.1$ barns, any mixture of these three elements at the stated number density gives approximately $2.39$–$3.48\%$. This is a bound within the idealized mixture model, not a claim that every such composition is stable austenite.

Take $\varphi (p_1)+\bar{\varphi }(p_2)\to \varphi (p_3)+\bar{\varphi }(p_4)$ with the Mandelstam variables of i. The allowed tree-level Feynman diagram is the crossed $u$-channel $\Phi$ exchange in panel (c) of b's figure. At one end the incoming particle and outgoing antiparticle transfer charge $+2$; the other end absorbs that charge. The zero-charge $s$- and $t$-channel assignments are forbidden by the interaction vertices. ThusIn the center-of-mass frame with $|\mathbf{p}|\ll m$, $s=4m^2+O(|\mathbf{p}{|}^2)$ and $u=O(|\mathbf{p}{|}^2)$. For $M\gg m$, both scattering amplitudes approach $g^2/M^2$. The elastic two-body Lorentz-invariant phase space and incident flux agree for the two experiments. Usingfor distinguishable final particles, and an extra factor $1/2!$ when integrating the identical-particle final state over the whole sphere, givesThere is no additional division for identical incoming particles in the definition of a beam scattering cross-section. Before taking $M/m\to \infty$, the zero-momentum ratio is $\frac{1}{2}[M^2/(M^2-4m^2){]}^2$, provided the threshold is away from the heavy pole.

Near $s=M^2$, the bare $s$-channel scalar propagator makes fixed-order perturbation theory fail: repeated self-energy insertions contain powers of ${\Sigma }_{\Phi }(s)/(s-M^2)$. The $\Phi \to \varphi \varphi$ channel is open for $M>2m$, giving the leading decay widthResumming the self-energy gives a finite Breit-Wigner resonance cross section, with denominator $s-M_{\mathrm{pole}}^2+i{M}_{\mathrm{pole}}{\Gamma }_{\Phi }$ near the pole, rather than an infinite scattering cross-section. Also, $s\simeq M^2$ with $M\gg m$ lies outside the assumed nonrelativistic incoming kinematics; the ratio $1/2$ is not a resonance prediction.

Label the incoming four-momenta $p_1,p_2$ and the outgoing ones $p_3,p_4$. Define the Mandelstam variables $s=(p_1+p_2{)}^2$, $t=(p_1-p_3{)}^2$, $u=(p_1-p_4{)}^2$. Two incoming $\varphi$ particles have conserved charge $+2$, so the only tree-level Feynman diagram is an $s$-channel $\Phi$ exchange, shown in panel (b) of b's figure. Each end converts two charge-one lines into one charge-two line. There are no $t$- or $u$-channel exchanges: an incoming and an outgoing particle at such an end would transfer zero charge, for which this theory has no cubic interaction vertex.The single Feynman diagram is already symmetric under exchanging the two outgoing particles. This does not add another copy of the scattering amplitude; the identical final-state symmetry factor enters the integrated scattering cross-section instead.

A scattering phase shift measures the change of a partial wave's outgoing phase relative to the free radial solution. For a real central potential and elastic scattering, probability conservation gives a unit-modulus Partial-wave S-matrix $S_l=e^{2i{\delta }_l}$. Indeedso the incoming amplitude is unchanged and the outgoing one acquires $S_l$.

Expanding an incident plane wave in Legendre polynomials and comparing the outgoing spherical wave givesUse $d\sigma /d\Omega =|f{|}^2$ and $\int d\Omega P_l{P}_{l^{\prime }}=4\pi {\delta }_{l{l}^{\prime }}/(2l+1)$ to remove cross terms:This is the elastic total scattering cross-section for the one-particle central-potential problem, with no identical-particle symmetrization factor.

In the forward direction ${\mathbf{k}}_f={\mathbf{k}}_i$, the first Born amplitude $-C\int V$ is real. Symmetrize the second-order double integral under $\mathbf{r}\leftrightarrow {\mathbf{r}}^{\prime }$ before taking its imaginary part. This replaces $e^{-i{\mathbf{k}}_i\cdot \mathbf{d}}$ by $\cos ({\mathbf{k}}_i\cdot \mathbf{d})$, and givesThe preceding total-cross-section integral equals the same real integral multiplied by $4\pi /k$. ThereforeThis is the optical theorem at second Born order. Comparing only the real first-order forward amplitude with a nonzero second-order scattering cross-section would mix perturbative orders. Terms of order $V^3$ require both the next amplitude order and interference in the scattering cross-section.

Define $A=m{g}^2/(2\pi {\hslash }^2)$, which has units of energy times length. Substituting the result of (b) into (c) givesAbsorb the real cutoff shift into the physical detuning,It follows thatUsing $k=\sqrt{2m{E}_k}/\hslash$, identifyHere $\gamma$ has units of square root of energy. This is the small-cutoff, renormalized resonant-level amplitude; terms vanishing with $k\delta$ have been omitted. Taking $\delta \to 0$ at fixed measured detuning requires the bare ${ϵ}_0$ to depend on the cutoff.

The sign is checked by $f_k^{-1}=-(E_k-{\widetilde{ϵ}}_0)/A-ik$, which has the elastic $s$-wave unitarity form. Thus $\mathrm{Im}{f}_k=k|f_k{|}^2$, consistent with the optical theorem and ${\sigma }_{\mathrm{tot}}=4\pi |f_k{|}^2$. If the renormalized level is positive, at $E_k={\widetilde{ϵ}}_0$ the amplitude is $i/k$ and the scattering cross-section reaches $4\pi /k^2$.

Outside a short-range scattering region, the stationary Schrödinger equation reduces to the two-dimensional Helmholtz equation. Separation in polar coordinates gives outgoing radial Hankel functions, whose large-$r$ dependence is $r^{-1/2}{e}^{ikr}$. The same power follows from conserved probability current: a circle has length proportional to $r$, so the squared outgoing amplitude must decrease as $1/r$.

For an incident plane wave in the $x$ direction, adopt the dimensionless amplitude conventionThe phase $e^{i\pi /4}$ and factor $k^{-1/2}$ specify the definition of $f$; they can be absorbed into a differently normalized scattering amplitude. Here they are kept exactly so that the later two-dimensional scattering cross-section follows consistently.

The incident plane wave carries probability current $j_{\mathrm{in}}=\hslash k/m$. For the separated outgoing wave from the previous part,The rate through a distant arc is $r{j}_r^{\mathrm{sc}}d\theta$. Dividing by incident flux defines the differential two-dimensional scattering cross-section,Because $f$ is dimensionless, $\lambda$ has units of length, as appropriate for scattering in a plane.

The partial-wave scattering channel with orbital angular momentum quantum number one. Its angular dependence is built from degree-one spherical harmonics. For a finite-range potential away from a threshold pole, the finite-range partial-wave threshold law gives phase shift ${\delta }_1=O(k^3)$ and a partial elastic scattering cross-section proportional to $k^4$ as the relative wavenumber $k$ tends to zero.

The monostatic radar cross section is $4\pi$ times the backward differential scattering cross-section. Equivalently, ${\sigma }_{\mathrm{back}}={\lim }_{r\to \infty }4\pi r^2{S}_{\mathrm{back}}/S_{\mathrm{inc}}$. It is a directional scattering measure with dimensions of area, not generally the geometric area or the total scattering cross-section. The definition may additionally specify the transmitted and received polarizations.